Rogue Waves of the Kundu-DNLS Equation

Shibao Shan, Chuanzhong Li, Jingsong He

Department of Mathematics, Ningbo University, Ningbo, China.

DOI: 10.4236/ojapps.2013.31B1020                

Abstract

In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Furthermore, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed.

Keywords

Kundu-DNLS Equation; Darboux Transformation; Rogue Waves

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. S. Johnson, “On the Modulation of Water Waves in the Neighbourhood of kh 1.363,”Proceedings of the Royal Society A, Vol. 357, No. 1689,1977, pp. 131-141. doi：10.1098/rspa.1977.0159
[2]	S. W. Xu, J. S. He and L. H. Wang, “The Darboux Transformation of the Derivative Nonlinear Schrodinger Equation,” Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 30, 2011, p. 305203. doi：10.1088/1751-8113/44/30/305203
[3]	A. Kundu, “Integrable Hierarchy of Higher Nonlinear Schrodinger Type Equations, Symmetry, Integrability and Geometry,” Methods and Applications, Vol. 2, No. 12, 2006.
[4]	A. Kundu, “Landau-Lifshitz and Higher-Order Nonlinear Systems Gauge Generated from Nonlinear schr\"{o}dinger Type Equations,” Jour-nal of Mathematical Physics, Vol. 25, No. 12,1984, p. 3433. doi：10.1063/1.526113
[5]	G. Tan\v{o}glu, “Hirota Method for Solving Reaction-Diffusion Equations with Generalized Nonlinearity,” International Journal of Nonlinear Science, Vol. 1,2006, pp. 30-36.
[6]	M. J. Ablowitz and P. A. Clarkson and Solitons, “Nonlinear Evolution Equations and Inverse Scattering,” Journal of Fluid Mechanics, Vol. 244, 1992, pp.721-725.
[7]	S. Kakei, N. Sasa and J. Satsuma, “Bilinearization of a Generalized Derivative Nonlinear schr\"{o}dinger Equation,” Journal of the Physical Society of Japan, Vol. 64, 1995, pp.1519-1523. doi：10.1143/JPSJ.64.1519
[8]	J. S. He, L. Zhang, Y. Cheng and Y. S. Li, “Determinant representation of Darboux transformation for the AKNS System,” Science in China Series A: Mathematics, Vol. 49, No. 12, 2006, pp. 1867-1878. doi：10.1007/s11425-006-2025-1
[9]	C. Z. Li, J. S. He and K. Porsezian, “Rogue Waves of the Hirota and the Maxwell-Bloch Equation,” Physical Review E, Vol. 87, 2013, p. 012913. doi：10.1103/PhysRevE.87.012913
[10]	J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian and A. S. Fokas, “A Generating Mechanism for Higher Order Rogue Waves”.